A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension

Abstract

Let (X, S) be a set system on an n-point set X. The discrepancy of S is defined as the minimum of the largest deviation from an even split, over all subsets of S ∈ S and two-colorings χ on X. We consider the scenario where, for any subset X ′ ⊆ X of size m ≤ n and for any parameter 1 ≤ k ≤ m, the number of restrictions of the sets of S toIn this case we show that there exists a coloring χ with discrepancy bound O * (|S| 1/2−d1/(2d) n (d1−1)/(2d) ), for each S ∈ S, where O * (·) hides a polylogarithmic factor in n. This bound is tight up to a polylogarithmic factor [20,22] and the corresponding coloring χ can be computed in expected polynomial time using the very recent machinery of Lovett and Meka for constructive discrepancy minimization [19]. Our bound improves and generalizes the bounds obtained from the machinery of Har-Peled and Sharir [14] (and the follow-up work in [26]) for points and halfspaces in d-space for d ≥ 3.